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Questions tagged [arithmetic-progressions]

Questions related to arithmetic progressions, which are sequences of numbers such that the difference between consecutive terms is constant

An arithmetic progression or arithmetic sequence is a sequence of numbers such that the common difference between consecutive terms is constant. For instance, the sequence 15, 13, 11, 9, 7, $\ldots$ is an arithmetic progression with common difference –2.

If the first term of an arithmetic progression is $a_1$, and the common difference is $d$, then the $n$th term of the sequence $(a_n)$ is $$a_n = a_1 + (n-1)d.$$

If the common difference $d$ is—

  • positive, the terms increase to positive infinity.
  • negative, the terms decrease to negative infinity.

A finite portion of an arithmetic progression is called a finite arithmetic progression or sometimes just an arithmetic progression. The sum of a finite arithmetic progression is called an arithmetic series.

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Can the numbers $8, 15, 24$ be terms of an arithmetic progression with the common difference anything other than 1?

Let the common ratio/difference of the arithmetic progression be a number d. The exercise forces $d=/=1$ (d cannot be one). Is there any sort of proof for this exercise or am I supposed to play a guessing game? For example: We say $15=8+k*d$, where…
questionasker22
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An Arithmetic Progression Question

Let $(a_{n})_{n\geq 1}$ be an arithmetic progression with $r>0$ such as $\vee n\geq 1,\vee x\in (a_{n},a_{n+1})$ we have $\left [ x \right ]=\left [ a_{n} \right ]$. Prove that ∃m$\in N,m\neq 0$ such as $r=\frac{1}{m}$. I figured out that $x\in…
omega123
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Doubt on sequence

As I was reading a chapter sequence in maths then I come up with certain questions that What is sequence? Answer which I got on Google is something which goes repeating itself regularly. How many types of sequence are there? Answer which I…
user87284
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This is a matter of numerical sequences, it involves Arithmetic Progression.

Considering that the numerical sequence $a_n, n = 1, 2, 3, ...$ shown in table I is constructed by intercalating the terms of the three sequences presented in table II, judge the following items. The terms $3n - 1, n = 1, 2, 3, ...$ of the sequence…
funfun
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Analytic way to find geometric progression denominator.

The fourth, sixth, and fourteenth members of a variable arithmetic progression form a geometric progression. Find the denominator of this geometric progression. What is the most rational way?
Mouvre
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Partitioning integers to avoid arithmetic progressions of length $3$

Arithmetic Progressions, Quote: "A finite sequence like $\{10,13,16,19\}$. This is an arithmetic progression of length $4$, of (constant) difference $3$ and an initial value $10$." In this first finite sequence my guess is that length $4$ means the…
user366820
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Find the 9th element of the arithmetic progression given the sum of 3 times 3rd term, and 6 times 12th term

I'm trying to solve this sequence but I can't find anything related to my exact problem. Given the following: $3$a3 + $6$a12 = 81 Specifically, I'm trying to find the 9th element [a9] as well as d - common difference. I know of the formula an =…
Daniel
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Being simultaneously an arithmetic progression and a geometric one

Let $a,b,c$ be three consecutive terms of an arithmetic progression and a geometric one($a \neq 0$).Find the value of: $$\frac{a^3b+2b^2c^2-c^4}{a^2b^2+b^4-3c^4}$$ My attempts:Letting $a=b=c$ shows us that the answer is $-2$.Writing $a,b,c$ using…
Taha Akbari
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How do I answer this arithmetic sequence question?

Calculate the number of terms in the arithmetic sequence $$a, a + d, a + 2d,\cdots, a + (n-1)d.$$ I don't have a problem answering this question with an integer sequence but I'm a bit lost with what to do for the general formula.
Dan A
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If $a_1,a_2,a_3.......a_{2n+1}$ are in AP, then

Value of $$\frac{a_{2n+1}-a_1}{a_{2n+1}+a_1} + \frac{a_{2n}-a_2}{a_{2n}+a_2} + .... \frac{a_{n+2}-a_n}{a_{n+2}+a_n}$$ WHAT I DID The denominators will all be equal. So it’s simply addition of…
Aditya
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There are four terms in an AP such that their is sum is 50 and the greatest number is 4 times least, then the numbers are

Let the terms of the AP be $$a-3d, a-d, a+d, a+3d$$ So their addition will give 50 $$4a=50$$ This is where the whole thing breaks down. I realize that the fourth term is 4 times that of first. So some may say that fourth term will be $4a-12d$. But…
Aditya
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Arithmetic Progression Sum of Sequence

The sum of the first 19 terms of an arithmetic progression is equal to twice of the value of 10th term. The value of 10th term will be?
Hasaan Ahmed Khan
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If the sum of the first n terms of an Arithmetic Progression is equal to $n^2 + 3n$

If the sum of the first n terms of an Arithmetic Progression is equal to $n^2 + 3n$ then the first term of the Arithmetic Progression is I tried to solve this question by putting sum of of $n^th$ term of an Arithmetic Progression that is…
Ankit Kumar
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Arithmetic progression formula

To find out the sum of arithmetic progression it has been given S=n(a+l)/2 And this formula has been derived in this manner $$ S=a+(a+d)+(a+2d)+.........+(l-2d)+(l-d)+l $$ And by reversing $$ S=l+(l-d)+(l-2d)+........+(a+2d)+(a+d)+a $$ On adding…
user230507
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Arithmetic Progressions of three squares

I was reading a pdf (Arithmetic Progressions of Three Squares by Keith Conrad) and I have a question about Theorem 3.5. It says that we can use Dirichlet's theorem on primes in arithmetic progression to prove the Theorem, but the proof is on pages…
Dr.Mathematics
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